PV ( Present Value)  

Present Value PV:  Assuming that $1,000  tomorrow will have more value to you than $1000 in 5 years we need to find a way to compute what the actual value of $1,000 in 5 years is to us now.  The formula is

PV = FV ( Future Value ) /  ( 1 + Interest ) ^ n

To make the calculation simple let’s test the value in 3 years with a 4% interest

PV = $1000 / ( 1 + 0.04 ) ^ 3 = $1000 / 1.04 ^3 = $1000 / 1.12 = $892

We can calculate the PV of 1000 a year from now, add to the PV of $1000 two years from now and add it up to the value of $1000 three years from now   – A more elegant formula is PV=∑ FV / (1+r)^t  for t=1 to n

Example we have to choose between $1000 now and $200 a year for 6 years starting now:

	Option 1 Today	$    1,000
Year	0	1	2	3	4	5	6
	Option 2  Cash Flow	$        200	$        200	$        200	$        200	$        200	$        200
	Discount Rate	12.0%
	Discounted Cash Flow	$        200	$        179	$        159	$        142	$        127	$        113
	PV	$        921

 ( Net Present Value )

The net present value is simply the present value of all future cash flows, discounted back to the present time at the appropriate discount rate, deducting the cost to acquire those cash flows. In other words, NPV is simply the value now minus the cost now.

Net present value (NPV) is determined by calculating the costs (negative cash flows) and benefits (positive cash flows) for each period of an investment, discounted back to the present time at the appropriate discount rate. The NPV is the sum of all the discounted future cash flows.

The NPV function in which the first payment is made at the end of the first period. Therefore, this value is included as the first value1 argument to the NPV function.

NPV= ∑ { Cash Flow / (1+r)^t} 

The NPV function in which the first payment is made at the start of the first period.

NPV= ∑ { Cash Flow / (1+r)^t} – Initial Investment

𝑁𝑃𝑉 = 𝑃𝑉 (𝐵𝑒𝑛𝑒𝑓𝑖𝑡𝑠) − 𝑃𝑉(𝐶𝑜𝑠𝑡𝑠)

𝑁𝑃𝑉 = 𝑃𝑉(𝐴𝑙𝑙 𝑝𝑟𝑜𝑗𝑒𝑐𝑡 𝑐𝑎𝑠ℎ 𝑓𝑙𝑜𝑤𝑠)

There are 3 categories NPV will fall into:

Positive NPV. If NPV is positive then it means you’re paying less than what the asset is worth and signifies a project that has net positive benefits after accounting for the time value of money
Negative NPV. If NPV is negative then it means that you’re paying more than what the asset is worth and indicates the project will result in a loss for the investor
Zero NPV. If NPV is zero then it means you’re paying exactly what the asset is worth at that assumed rate of return.

We are proposed to invest $1000 and receive $500 for 3 years. The Discounted Rate is 5% – Is this a good deal ?

Example Below  After the 1^st year we calculate the discounted cash flow

Year	Cash Flow	Disc Cash flow	Rate	5%
0	$               (1,000)	$         (1,000)
1	$                    500	$                  476	=E5/(1+5%)^D5
2	$                    500	$                  454	=E6/(1+5%)^D6
3	$                    500	$                  432
NPV	Sum=$500	$                  362

Payback Time

A project requires an initial investment of $3,000 upfront. It will produce an annual income of $500 every year for 6 years. What is this project’s payback period without considering discounting?

Year 1 = $500    Total = $500

Year 2 = $500     Total = $1000

Year 3 = $500   Total = $1500

Year 4 = $500  Total = $2000  Payback Period is 2 years

A project requires an initial investment of $2,000 upfront. It will produce an annual income of $1,000 every year for 3 years. Assume a discount rate of 10%. What is the above project’s discounted payback period?

Discounted year 1 CF = 1000/1.1=909

Discounted year 2 CF = 1000/1.1^2=826

Discounted year 3 CF = 1000/1.1^3=751

2000 – 909 – 826 = 264

264/ 751 = 0.352

2+0.352=$2.352